Interdisciplinary Applied Mathematics

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The procedure for distributed parameter modeling is given by:

1. A quasi-one-dimensional modeling of the system is first performed, where only the displacements associated with the dominant modes are considered, while the ones in the other directions are neglected. This reduces the dependence of the system behavior to a single coordinate.

2. The characteristic equations are derived using modal analysis, which is described in detail later in this section.

3. Modal analysis techniques are used to find the solution to the governing equations in terms of normalized mode shapes and generalized coordinates by the use of the mode-superposition principle.

4. A Galerkin-like approach is then used to generate an infinite set of uncoupled ODEs that represent the system by means of an infinite number of single-degree-of-freedom lumped-parameter systems.

5. The characteristic equations of the system that describe the linear relations between incremental variations of the port variables around a    stable    bias    point    are derived.    The    equations    are    then    linearized

around the bias point.

6. Using the possible characteristic equations and the numerous equivalent circuit representations, a circuit representation that represents the practical situation in the most appropriate way is selected.

The steps in the distributed parameter approach are similar to those in the lumped parameter approach, except that the continuous system is modeled using modal analysis in the distributed parameter approach to reduce the degrees of freedom. We consider the actuator shown in Figure 17.5 to illustrate the distributed parameter approach. The electrical and mechanical domains are coupled through the surface of the flexible beam, which can be thought of as an infinite number of localized individual electrostatic transducers. The electrical energy stored in each such element of infinitesimal area SA is given by

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